Fixed-point property

A mathematical object X has the fixed-point property if every suitably well-behaved mapping from X to itself has a fixed point. The term is most commonly used to describe topological spaces on which every continuous mapping has a fixed point. But another use is in order theory, where a partially ordered set P is said to have the fixed point property if every increasing function on P has a fixed point. Let A be an object in the C. Then A has the fixed-point property if every morphism (i.e., every function) has a fixed point. The most common usage is when C = Top is the . Then a topological space X has the fixed-point property if every continuous map has a fixed point. In the , the objects with the fixed-point property are precisely the singletons. The closed interval [0,1] has the fixed point property: Let f: [0,1] → [0,1] be a continuous mapping. If f(0) = 0 or f(1) = 1, then our mapping has a fixed point at 0 or 1. If not, then f(0) > 0 and f(1) − 1 < 0. Thus the function g(x) = f(x) − x is a continuous real valued function which is positive at x = 0 and negative at x = 1. By the intermediate value theorem, there is some point x0 with g(x0) = 0, which is to say that f(x0) − x0 = 0, and so x0 is a fixed point. The open interval does not have the fixed-point property. The mapping f(x) = x2 has no fixed point on the interval (0,1). The closed interval is a special case of the closed disc, which in any finite dimension has the fixed-point property by the Brouwer fixed-point theorem. A retract A of a space X with the fixed-point property also has the fixed-point property. This is because if is a retraction and is any continuous function, then the composition (where is inclusion) has a fixed point. That is, there is such that . Since we have that and therefore A topological space has the fixed-point property if and only if its identity map is universal. A product of spaces with the fixed-point property in general fails to have the fixed-point property even if one of the spaces is the closed real interval.

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MATH-416: Abstract analysis on groups

We study analytic phenomena on groups, notably paradoxical decompositions, fixed point properties and harmonic functions.

Invariant integrals on topological groups

Vasco Schiavo

We generalize the fixed-point property for discrete groups acting on convex cones given by Monod in [23] to topological groups. At first, we focus on describing this fixed-point property from a functional point of view, and then we look at the class of groups that have it. Finally, we go through some applications of this fixed-point property. To accomplish these tasks, we introduce a new class of normed Riesz spaces that depend on group representation. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

ACADEMIC PRESS INC ELSEVIER SCIENCE2022

Invariant Integrals on Topological Groups and Applications

Vasco Schiavo

We study a fixed point property for linear actions of discrete groups on weakly complete convex proper cones in locally convex topological vector spaces. We search to understand the class of discrete groups which enjoys this property and we try to generalize it for Hausdorff topological groups.

EPFL2021

Representing groups against all odds

Maxime Gheysens

We investigate how probability tools can be useful to study representations of non-amenable groups. A suitable notion of "probabilistic subgroup" is proposed for locally compact groups, and is valuable to induction of representations. Nonamenable groups admit nonabelian free subgroups in that measure-theoretical sense. Consequences for affine actions and for unitarizability are then drawn. In particular, we obtain a new characterization of amenability via some affine actions on Hilbert spaces. Along the way, various fixed-point properties for groups are studied. We also give a survey of several useful facts about group representations on Banach spaces, continuity of group actions, compactness of convex hulls in locally convex spaces, and measurability pathologies in Banach spaces.

EPFL2017